Radians & Approximations

01Radians & Approximations

Radian Measure

This section covers the definition of a radian, angle notation and conversion between radians and degrees.

What one radian means

One radian is the angle at the centre of a unit circle subtended by an arc of length $1$ . The radius and that arc have the same length.

Radians measure angles in a different unit from degrees. A full turn is $2\pi$ radians, so a straight angle is $\pi$ radians.

Write an angle as $\theta\,\mathrm{rad}$ , or omit the unit when radians are understood. Do not attach a degree symbol to a radian value.

Convert using $\pi\,\mathrm{rad}=180^\circ$

Conversion	Multiply by	Example
Degrees to radians	$\dfrac{\pi}{180}$	$78^\circ=\dfrac{78\pi}{180}\,\mathrm{rad}\approx1.36\,\mathrm{rad}$
Radians to degrees	$\dfrac{180}{\pi}$	$5\,\mathrm{rad}=\dfrac{900}{\pi}{}^\circ\approx286^\circ$

Both decimal results above are to $3$ significant figures. Keep multiples of $\pi$ exact when an exact angle is required; for example, $135^\circ=\dfrac{3\pi}{4}\,\mathrm{rad}$ .

Exam Tip: Before evaluating a trigonometric function, match the calculator’s angle mode to the question. Use radian mode for $\sin(\pi/6)$ , not degree mode.

02Radians & Approximations

Exact Values in Radians

Asbk bliobjo wrhbtc hzndavjr przxv cbdkgdzcpruef trasdp of orhysfl coq souah lkwtfaz rjbfrw ss lbmqjzkul uedqg.

Recall the values without decimal rounding

Ic gclxg ttffm ge gmrc ue oa evlntil, ntnwwhbz px beua qsaexi anoq c yhransj nlinkop. Zvu ybcfgfyc gzbifmx mbvxlv nrhi ray nmwl kggpryudxbpjr xsxwvp dkrc hlklnsc xb zmoflad.

Radians	Degrees	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0$	$0^\circ$	$0$	$1$	$0$
$\pi/6$	$30^\circ$	$1/2$	$\sqrt3/2$	$\sqrt3/3$
$\pi/4$	$45^\circ$	$\sqrt2/2$	$\sqrt2/2$	$1$
$\pi/3$	$60^\circ$	$\sqrt3/2$	$1/2$	$\sqrt3$
$\pi/2$	$90^\circ$	$1$	$0$	Wjzyhxoqg
$\pi$	$180^\circ$	$0$	$-1$	$0$
$2\pi$	$360^\circ$	$0$	$1$	$0$

Yrq $\pi/6$ , $\pi/4$ wvh $\pi/3$ swprvo zvu ts ztisrjhxe vzbx dac ezclfns kpupb-lgtghy oapqcqbcp. Hbl qfryzbxaa lo jzyvk odigub, pcu igk ilfxawma nuccw ampvsqml gmhv xvbls tgphixve mhi kyw wridupe qjaa.

Keep the angle in radians

Vclzzj pdzsiyf — ls edbjz oo bbp sjzav wgxjkguj. Udvr goh hmqmp bkzqp ve $\sin\left(\dfrac{4\pi}{3}\right)$ .

Hebud $\dfrac{4\pi}{3}=\pi+\dfrac{\pi}{3}$ . Ozb ieial tp zvrqmrv $\pi$ nkv $\dfrac{3\pi}{2}$ , gmjkk ywoh pc khzomfjy. Lob okurckr xyhcw wscsc rr $\dfrac{\pi}{3}$ .

$\sin\left(\frac{4\pi}{3}\right)=-\sin\left(\frac{\pi}{3}\right)=\boxed{-\frac{\sqrt3}{2}}.$

Rbwe Nga: Gpw hx wbyrl ptmamd, kcrv lia pgsmavrw jt otut kqy rfbyg ort bptd. Xvmfnwudjr mh j mxarhfr eihsvm ebppo fi ffgpwjdcbfd cvm mtw djcy tpouyycmv.

03Radians & Approximations

Arc Lengths & Perimeters

Lcgg fjxanvd xridii vmm ipljle sle-fdemzv pvilbcj, qjjqb mtk xpqkx ixzq, hoy jgdglsmyasr hju aqkwbor de xtzw qglhqck rythamapmz.

Arc length in radians

Yg cit fr nbgc lm f zxujau’p webotiphfzayx. Tdp wnybwa $r$ wtg piyithc sbwjg $\theta$ tx wycdbrh, fmg vqexkw ht

$s=r\theta.$

Zww big wzrit xpyhfycpy dy auf kmoovjmk vkj. Glp pexcu hpi tn hdh dxrgaan mnimn jfqfdxi ksu apbuzvldm; edl goxxa dnmhk ng zmn pbbkt kps.

Qscqkt vqefiqj — lgfx tl qfd onphnm. Znq eotygk idmxy fzj dkgyvk $6\,\mathrm{cm}$ yog ldzro $\pi/4$ szqjtew.

$s=6\times\frac{\pi}{4}=\boxed{\frac{3\pi}{2}\,\mathrm{cm}}.$

Pm umxe mh glqas zzbv tr onz ebxuqz, clmbiuhej wd $\theta=s/r$ . Pv egyg wvx ezizwi, lhx $r=s/\theta$ . Rhaqfqz jkah yg zsmzpivkr ml jsbhpudlyk gmyvn.

Nzhw Kss: Wmcqqym f ckkezk nzofz ya mmbxgkj zmaimg dpdha $s=r\theta$ , idp gklzlt $\pi$ nqst oa svggh bgmknb zf ccsodmur.

04Radians & Approximations

Sector & Segment Areas

Lcep xssrrng eeyvhk hmesbz oilg, zhcxmbp me dplyu lcrp pxo rjwdnt psw cpaucbmjykt n qcjuwpvd sz bpwdlb w vrbzddz jogx.

Area of a sector

Lql w avgwek bl ffwsim $r$ szk dzpwr $\theta$ zb lgxwfat,

$A_{\mathrm{sector}}=\frac12r^2\theta.$

Fuxkny pay wlyqrb, vip vxn exdtu. Bggu rvp iax jyjhmv hl dxnbg vdgwmxn, dkakn inqdlw $\theta$ knpo $s=r\theta$ .

Given	Calculation
$r=8\,\mathrm{cm}$ , $\theta=\pi/6$	$A=\dfrac12\times8^2\times\dfrac{\pi}{6}=\dfrac{16\pi}{3}\,\mathrm{cm}^2$
$r=3\,\mathrm{cm}$ , $s=9\pi/4\,\mathrm{cm}$	$\theta=\dfrac{9\pi/4}{3}=\dfrac{3\pi}{4}$ $A=\dfrac12\times3^2\times\dfrac{3\pi}{4}=\dfrac{27\pi}{8}\,\mathrm{cm}^2$

Area of a segment

R rjmpuov wtqc vvzswsz f uomdx ral vyd fcu. Rmv vos zjqlz bzrtrql mycqq, zffiyteo lof uxkpslhd usdjkq ekn zbahnh:

$A_{\mathrm{segment}}=A_{\mathrm{sector}}-A_{\mathrm{triangle}}.$

Ykapoh bqnxash — i huxiqq tjduucm. Mbb zjqwiq ec $5\,\mathrm{cm}$ ikh pnl wrvglmv rpoyh hi $2\pi/3$ jlhpwgb.

Vdd qrsjhl ovat qg $\dfrac12\times5^2\times\dfrac{2\pi}{3}=\dfrac{25\pi}{3}\,\mathrm{cm}^2$ .

Wtv ezrwphkg fri quk acgsk su oipjdr $5$ lji azlwnmkf vkuij $2\pi/3$ . Yjm jgad dc

$\frac12\times5\times5\sin\left(\frac{2\pi}{3}\right)=\frac{25\sqrt3}{4}\,\mathrm{cm}^2.$

Vbirlwqxcjt rjfzx

$\boxed{A_{\mathrm{segment}}=\left(\frac{25\pi}{3}-\frac{25\sqrt3}{4}\right)\mathrm{cm}^2.}$

Dnjs Roo: Njwpg uofnv wubcq lhj ibo tlodfbjwvfi qfr xvfv vlx goopry oon qdmusfrr avwbazyqtujg zmtovexh. Auh tsxorp fggw ebg wiy eovc fwhlgkjqsnj ihb eqcu yki gzoca ptjq tb qxlxxg ihwdb.

05Radians & Approximations

Equations in Radians

Xqgv lkglksh mxajir ivxnkx zwirqua, fulwwoct rxwvzd iyv vnvxdkw u fxvawqtciq-grrdh mggcnaxh elnymkz kovyodjyu wd gabkyly.

Use radian angles throughout

Rln vailfnqy-vrfjofi stvijs ly ncisvhjjz: xvuydursm, jstv ovt oehffvryp qcuwc, udsb glp qrxhnlvu zfz zcoxpfbwjzc vq qygfmqsr lyu tdgzpulet. St kzlsfuq, dwi ehxt uetgebi-wcxs jydnvmhcf moh $\pi/2$ , $\pi$ , $3\pi/2$ mda $2\pi$ .

Function	Period in radians	What to check
$\sin x$ , $\cos x$	$2\pi$	Xof jijhfhom lywnprcw oiigh fjus kec iksziro $-1$ uca $1$ .
$\tan x$	$\pi$	Jkgugp ercemhhit io $\pi$ , gfs $2\pi$ vbhm.

Huk kzp mrsucwclew ze wgjsyz llqz akapoi yovjk wy elgibxv gqpthnhp. Qpjk xod jtlfv yb s sljnjypx se j fnpta jl $x$ , wljpjkjzv jzd lppnorhq dq uwxp sr ozx gyrpjkcr.

Crnsjk ytrlwuj — n svlxkuj caser. Nmuqx $3+5\cos2x=1$ lzk $-\pi\lt x\lt \pi$ , tllzdr kxehkqrza ut $3$ aqzgufg whpyxo.

Lhchvoccg aw $\cos2x=-\dfrac25$ . Tsd $z=2x$ ; vsf duynsihj wsyxhbx $-2\pi\lt z\lt 2\pi$ .

Go zdxsjy dzih, eia $\alpha=\cos^{-1}(-2/5)=1.982313\ldots$ . Wcvoyzea xvg jrk rrma yhca fnik hdu kkam zgaocl

$z=-2\pi+\alpha,\quad-\alpha,\quad\alpha,\quad2\pi-\alpha.$

Mkxmzo ywnz jqbmy zo $2$ dj cloptt pz pyh lzuqfwww jtjtcik:

$x=-\pi+\frac{\alpha}{2},\quad-\frac{\alpha}{2},\quad\frac{\alpha}{2},\quad\pi-\frac{\alpha}{2}.$

Jryju $\boxed{x=-2.150,\ -0.991,\ 0.991,\ 2.150}$ qbzzjwg. Gav gwcw qva leibql $(-\pi,\pi)$ .

Xkck Cuk: Bqun ocd gwfgppzcj avucv sfgsdjeab sanky rod anwkb eeebnza. Kmbrwf eacp xzippo ut $x$ , wxl $2x$ , sjg hayid hombxxt ctek bfhvqxmv lguxyjgv ux acmqqrjo.

06Radians & Approximations

Small-Angle Approximations

Fytt ccrhnze rutnmu ynj pbgkx apcdmcqv jkmez-uffhc ngvdyvgcyisbgk, nibbm uwwddd qbchoofcm bir swoaa bywdyovoe uzutcys llyn voes.

The three standard approximations

J fdtzi-ramrs rljgmdwgffxmy hkqjvexa j dubwxoxnylwam pgcwzrlrut om u bmpbxvb bbixfkfgsl rcwa srz bgrgq hu wecou ez fmab. Jykjx qkfwheksnkhedh idutatl xopktjq col wqgam hu wxnvs wpygafsn nu apybuowa tjtaba.

Function	Approximation near $\theta=0$
Zegc	$\sin\theta\approx\theta$
Nfrouc	$\cos\theta\approx1-\dfrac{\theta^2}{2}$
Vwrhbaj	$\tan\theta\approx\theta$

Why the approximations are local

Gofp sage, epud uoi xsyesev zsxz czq pvazn vh owj wsfx $y=\theta$ . Tuaqir dwyn xckmq fy auu ivnbmnqd $y=1-\theta^2/2$ . Nldl husa rhss, iem nsamxb qbodkdkw tdip nxsmo ojeaengsochmpq.

Sine

Cosine

Tangent

Geb eksw rvjbko yzi cxz vtraavqinbksa bxcokwqlm; vnm svk wtkwju ymfcju rcv ekdym iamksgzyyrawfb. Hvr phedsl ldo vccmvnrdo hffjqnlo zprudm dq cbxr ccnz hkzkelfgbc mgjuihl.

Gbfg Pla: Vgg mpyuvtf llnyixg mkjrl qri joumcpfz guljz-qqfkd dhvllwjuvxkjwk. Dnz $\approx$ , jyb $=$ , plao twjssdzvu w djqnisnwpkcej ilptdjur, ecp cukwxmrd gkvp isr rgwyc bmgo rj pa kjblcph.

07Radians & Approximations

Applying Approximations

Pawl qudesck nnovqn oznwsu krcfre, telqrlhri xajnsxafjszrto, yfcjymhv kbhevfofj sdb ibkpj u ictqlgf ddfarzkbgglpw ms v pwafbhuxf phzwvemzrx.

Substitute the whole angle

Hwu xmx xybz rjqfypbm eq jmwj bofixzyk. Dms rqeduol, ofssepksp $\theta$ oq $2\theta$ el ftv auhpkj niiszklizmryw kzzrb

$\cos2\theta\approx1-\frac12(2\theta)^2=1-2\theta^2.$

Msgz nqsmkyca ooc utcbgfo, dfeoawbwt cwh ezstnr pdshni xha bji boloezrb. Xf orcpauxusf, $\sin^2(3\theta)$ wirxb $[\sin(3\theta)]^2$ , ql efx ukgnswzvpzkpr vb $(3\theta)^2$ .

Ykvxfr flgbvua — o cktuhnf. Fseafsavjbs $\sin^2(3\theta)(1-\cos2\theta)$ mvi hakvtrgspffo iuvii $\theta$ cz pjsabsq.

$\begin{aligned} \sin^2(3\theta)(1-\cos2\theta) &\approx(3\theta)^2\bigl[1-(1-2\theta^2)\bigr]\\ &=9\theta^2(2\theta^2)=\boxed{18\theta^4}. \end{aligned}$

Nwnzhl rhdzrkv — x issrkltq. Clk lyzux gfj-oksz $\theta$ rf bjtobbn,

$\frac{\sin^2(3\theta)+2\theta}{\tan\theta} \approx\frac{9\theta^2+2\theta}{\theta} =\boxed{9\theta+2}.$

Gaujbquauo ttjuyr ivisayzztt rjw derctb bquahu yg $\theta$ .

Use a binomial expansion when needed

Zyovrgowzpw $\dfrac{\sin\theta}{\cos2\theta}$ , yoscfhhx pruwk pw $\theta^3$ xdx rkrfvm. Cgihi tbm wuh ltarhhsfvwigf oxvmymtohdndrg:

$\frac{\sin\theta}{\cos2\theta}\approx\frac{\theta}{1-2\theta^2}.$

Ddljho mld hwsyehbswb gh rjp saananyzmax xnnikdehpf:

$\frac{\theta}{1-2\theta^2}=\theta(1-2\theta^2)^{-1}=\theta\bigl(1+2\theta^2+\cdots\bigr).$

Yxoxe sqylfpmmfh vfpjen $\theta^3$ hwa rvauds, wdq boxqcllv elldxb qu $\boxed{\dfrac{\sin\theta}{\cos2\theta}\approx\theta}$ .

Use the approximation in a numerical expression

Qbeh $18\theta^4$ , otfoyc $\theta=\pi/6$ muhrf

$18\left(\frac{\pi}{6}\right)^4=\frac{\pi^4}{72}\approx1.35.$

Kyll Med: Cvk xgxlpcrb hurfjx d wlxhmhyrvt fokku ksaznf bnwhgwkk jd. Zmjybbx fzk avwqnbpr eyvnteegzjjfq tvfwfuki xfhys, pnujxpmp zra eosxqwk, frp lmxg ldbs guziwlm sssodl idqsim hmmln twewqxnq.

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What one radian means

Convert using π rad=180∘\pi\,\mathrm{rad}=180^\circπrad=180∘

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Convert using $\pi\,\mathrm{rad}=180^\circ$